There are actually plenty of animals and plants that undergo what is called parthenogenesis, in which the female essentially clones herself by exactly the way you describe–two of her eggs fusing and forming an embryo. Sometimes, this is actually controlled by bacteria that infect the females. This happens with some wasps, for example!
While you could imagine this happening in a human, our eggs have lots and lots of control mechanisms to prevent this from happening. But if you were to artificially inject a nucleus from one egg cell into another, you could get a viable zygote that fully develops into an embryo. That is how we clone mammals today.
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